Date of Award
Doctor of Philosophy (PhD)
Stuart, Steven J
In the last decade great efforts have been made to efficiently study the behaviors of rare-event systems. The methods and theories developed in pursuit of this goal are far from unified and are instead applied on a case-by-case basis where one makes a best educated guess as to which approach may afford the greatest chance of studying the rare-event system in question. The work contained here is directed at further development of one of these sets of methods and theories. Specifically, a new theory of dynamics in the Langevin path space is developed with emphasis on generating and sampling Langevin trajectories that exhibit rare transitions.
The quest to formulate path dynamics or path sampling is not new. Rather, this work offers a new formulation of an old idea while keeping in mind two key issues: first, whatever theory and methods are developed they should be accessible to those simply seeking methods and easy to implement, since the goal is application to complex, large dimensional systems. Second, the rigorous limits of the problem should never be overlooked and one`s intuition should always be held in check by these limits of rigor.
Past formulations of techniques that employ a Langevin measure to either uncover very probable paths or to define path-wise dynamics tend to rely too heavily on intuition, allowing one to unknowingly overstep the rigorous limit of what can be done or said. With this in mind, a new Langevin path probability is derived once it is established that the more common representation can only be written without justification. The new Langevin measure is then used to demonstrate consequences related to applications that are not limited to a special function space; the space of functions that solve the given dynamical equation. This is simply a demonstration of uniqueness for Langevin`s equation.
Because working in the configuration or phase space is a familiar thing, path sampling and path dynamics are usually formulated on one of those spaces. Since the first part of this work revolves around the idea that the domain of Langevin trajectories is in neither of those spaces, an alternate formulation of dynamics in the Langevin path space is given. In this formulation the intuition developed by the regular dynamics in phase space is pushed aside for a different perspective. This forces the domain of the path dynamics to stay on the random force space, the independant variable of the Langevin trajectory. Once the path dynamics are formulated the scheme is demonstrated on a test potential and then used to close a fullerene at low temperature.
This work is unique amoung the literature in that it avoids forcing physical intuition from one problem on the study of another. In the end, the tools developed here are easy to understand and straightforward to implement. A number of the difficulties of the usual path sampling formulations are even avoided. For instance, one is not required to have access to a reactive trajectory a priori. Furthermore, since the domain of the problem is not forced onto the configuration space, the need for higher-order derivatives of the potential energy surface are avoided; only the gradient must exist. The main contributions of this work are an original method for the generation and sampling of reactive Langevin trajectories in rare-event systems and an alternative theory for understanding the Langevin path probability.
Dickson, Bradley, "PATH INTEGRAL FORMULATION OF LANGEINV DYNAMICS WITH APPLICATIONS TO RARE-EVENT SYSTEMS" (2006). All Dissertations. 24.